How to Laplace transform an equation?
2 Answers
Laplace transforms are a powerful mathematical tool used to convert differential equations into algebraic equations, which are often easier to solve. Here’s a step-by-step guide on how to apply the Laplace transform to an equation:
### 1. Understand the Laplace Transform
The Laplace transform of a function \( f(t) \) is given by:
\[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt \]
where:
- \( f(t) \) is the function of time \( t \),
- \( s \) is a complex number frequency parameter,
- \( F(s) \) is the Laplace transform of \( f(t) \).
### 2. Identify the Function and Differential Equation
Suppose you have a differential equation involving \( f(t) \) and its derivatives. For example, consider a simple first-order linear differential equation:
\[ \frac{dy(t)}{dt} + a y(t) = b \]
### 3. Take the Laplace Transform of Each Term
Apply the Laplace transform to both sides of the differential equation. Use the following properties:
- **Linearity**: \( \mathcal{L}\{a f(t) + b g(t)\} = a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)\} \)
- **Derivative**: \( \mathcal{L}\left\{\frac{d^n f(t)}{dt^n}\right\} = s^n \mathcal{L}\{f(t)\} - s^{n-1} f(0) - s^{n-2} \frac{df(0)}{dt} - \cdots - \frac{d^{n-1} f(0)}{dt^{n-1}} \)
For the given example:
1. **Apply the Laplace Transform**:
\[ \mathcal{L}\left\{\frac{dy(t)}{dt}\right\} + a \mathcal{L}\{y(t)\} = \mathcal{L}\{b\} \]
2. **Use the Derivative Property**:
\[ s Y(s) - y(0) + a Y(s) = \frac{b}{s} \]
where \( Y(s) = \mathcal{L}\{y(t)\} \).
### 4. Solve for the Laplace Transform of the Unknown Function
Rearrange the equation to solve for \( Y(s) \):
\[ (s + a) Y(s) - y(0) = \frac{b}{s} \]
\[ Y(s) = \frac{y(0) + \frac{b}{s}}{s + a} \]
### 5. Take the Inverse Laplace Transform
To find \( y(t) \), take the inverse Laplace transform of \( Y(s) \). Use known Laplace transform pairs and properties.
For the above \( Y(s) \), you can decompose it into simpler terms and apply inverse transforms:
\[ Y(s) = \frac{y(0)}{s + a} + \frac{b}{s(s + a)} \]
Find the inverse transform of each term:
- \( \mathcal{L}^{-1}\left\{\frac{y(0)}{s + a}\right\} = y(0) e^{-at} \)
- \( \mathcal{L}^{-1}\left\{\frac{b}{s(s + a)}\right\} \) requires partial fraction decomposition and then using known inverse transforms.
### 6. Combine the Results
Add the results from the inverse Laplace transforms to get the final solution:
\[ y(t) = y(0) e^{-at} + \text{(inverse of second term)} \]
### Summary
1. **Identify the function and its differential equation.**
2. **Apply the Laplace transform to each term.**
3. **Solve the resulting algebraic equation for the Laplace transform of the unknown function.**
4. **Use the inverse Laplace transform to find the solution in the time domain.**
By following these steps, you can effectively use the Laplace transform to solve differential equations and analyze systems in various applications.
### 1. Understand the Laplace Transform
The Laplace transform of a function \( f(t) \) is given by:
\[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt \]
where:
- \( f(t) \) is the function of time \( t \),
- \( s \) is a complex number frequency parameter,
- \( F(s) \) is the Laplace transform of \( f(t) \).
### 2. Identify the Function and Differential Equation
Suppose you have a differential equation involving \( f(t) \) and its derivatives. For example, consider a simple first-order linear differential equation:
\[ \frac{dy(t)}{dt} + a y(t) = b \]
### 3. Take the Laplace Transform of Each Term
Apply the Laplace transform to both sides of the differential equation. Use the following properties:
- **Linearity**: \( \mathcal{L}\{a f(t) + b g(t)\} = a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)\} \)
- **Derivative**: \( \mathcal{L}\left\{\frac{d^n f(t)}{dt^n}\right\} = s^n \mathcal{L}\{f(t)\} - s^{n-1} f(0) - s^{n-2} \frac{df(0)}{dt} - \cdots - \frac{d^{n-1} f(0)}{dt^{n-1}} \)
For the given example:
1. **Apply the Laplace Transform**:
\[ \mathcal{L}\left\{\frac{dy(t)}{dt}\right\} + a \mathcal{L}\{y(t)\} = \mathcal{L}\{b\} \]
2. **Use the Derivative Property**:
\[ s Y(s) - y(0) + a Y(s) = \frac{b}{s} \]
where \( Y(s) = \mathcal{L}\{y(t)\} \).
### 4. Solve for the Laplace Transform of the Unknown Function
Rearrange the equation to solve for \( Y(s) \):
\[ (s + a) Y(s) - y(0) = \frac{b}{s} \]
\[ Y(s) = \frac{y(0) + \frac{b}{s}}{s + a} \]
### 5. Take the Inverse Laplace Transform
To find \( y(t) \), take the inverse Laplace transform of \( Y(s) \). Use known Laplace transform pairs and properties.
For the above \( Y(s) \), you can decompose it into simpler terms and apply inverse transforms:
\[ Y(s) = \frac{y(0)}{s + a} + \frac{b}{s(s + a)} \]
Find the inverse transform of each term:
- \( \mathcal{L}^{-1}\left\{\frac{y(0)}{s + a}\right\} = y(0) e^{-at} \)
- \( \mathcal{L}^{-1}\left\{\frac{b}{s(s + a)}\right\} \) requires partial fraction decomposition and then using known inverse transforms.
### 6. Combine the Results
Add the results from the inverse Laplace transforms to get the final solution:
\[ y(t) = y(0) e^{-at} + \text{(inverse of second term)} \]
### Summary
1. **Identify the function and its differential equation.**
2. **Apply the Laplace transform to each term.**
3. **Solve the resulting algebraic equation for the Laplace transform of the unknown function.**
4. **Use the inverse Laplace transform to find the solution in the time domain.**
By following these steps, you can effectively use the Laplace transform to solve differential equations and analyze systems in various applications.